Snell / Morgan / Langford Elementary Analysis

The Commonwealth and International Library: Mathematics Division, Volume 2
1. Auflage 2014
ISBN: 978-1-4831-5898-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 304 Seiten, Web PDF

Elementary Analysis, Volume 2 introduces several of the ideas of modern mathematics in a casual manner and provides the practical experience in algebraic and analytic operations that lays a sound foundation of basic skills. This book focuses on the nature of number, algebraic and logical structure, groups, rings, fields, vector spaces, matrices, sequences, limits, functions and inverse functions, complex numbers, and probability. The logical structure of analysis given through the treatment of differentiation and integration, with applications to the trigonometric and logarithmic functions, is also briefly discussed. This volume begins with a description of the trigonometric functions of the general angle and an introduction to the binomial theorem and series. The rest of the chapters cover the numerical solution of equations, analytical geometry, Argand Diagram, numerical methods, and methods of approximation that form an important section of modern applied mathematics. This publication is valuable to teachers and students in training colleges.

Autoren/Hrsg.

Snell, K. S.

Morgan, J. B.

Langford, W. J.

Maxwell, E. A.

Inhaltsverzeichnis

1;Front Cover;1
2;Elementary Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;INTRODUCTION: A SUMMARY OF BASIC IDEAS FROM VOLUME 1;10
7;CHAPTER 15. TRIGONOMETRIC FUNCTIONS OF THE GENERAL ANGLE;16
7.1;Angles;16
7.2;Vectors;17
7.3;Unit vectors;17
7.4;Trigonometrical ratios;18
7.5;Sine and cosine;18
7.6;Special angles;19
7.7;The general angle;20
7.8;Periodic functions;22
7.9;Graphs of cos T and sin T;23
7.10;General formulae;24
7.11;The six trigonometrical functions;28
7.12;Graphs of tan . and cot .;29
7.13;Graphs of sec . and cosec .;30
7.14;General formulae;31
7.15;An important relation;33
7.16;Inequalities;36
8;CHAPTER 16. FUNCTIONS OF COMPOUND ANGLES;39
8.1;Unit vectors;39
8.2;Addition formulae;39
8.3;Multiple angles;41
8.4;Sums and products;42
8.5;Even and odd functions;47
8.6;Parameters and inverse functions;50
8.7;A useful transformation;53
9;CHAPTER 17. DIFFERENTIATION AND INTEGRATION OF THE TRIGONOMETRIC FUNCTIONS;58
9.1;Gradient of the sine function;58
9.2;Radians and degrees;60
9.3;TT radians = 180°.;61
9.4;A kinematical method;62
9.5;Differentiation of a vector;63
9.6;Differentiation of the unit vector;64
9.7;Motion in a circle;65
9.8;Some useful inequalities;67
9.9;Derivative of tan .;69
9.10;Summary;71
9.11;Integration of trigonometric functions;74
9.12;Integration by substitution;76
9.13;Polar coordinates;79
9.14;Area of a sector;81
10;CHAPTER 18. APPLICATIONS OF THE TRIGONOMETRIC FUNCTIONS;85
10.1;Inner product of vectors;85
10.2;Compound angle formulae;87
10.3;Triangle formulae;88
10.4;Triangle problems;93
10.5;Solution of triangles;94
10.6;Simple harmonic motion;99
10.7;Approximations;100
10.8;Three dimensions;103
10.9;Projections;105
11;CHAPTER 19. POLYNOMIALS;109
11.1;The remainder theorem;110
11.2;The factor theorem;112
11.3;The factorization theorem;113
11.4;The identity theorem;114
11.5;Synthetic division;118
11.6;Evaluation of a polynomial;120
11.7;The remainder theorem;121
11.8;The general polynomial;121
11.9;Repeated factors;124
12;CHAPTER 20. SYMMETRIC FUNCTIONS OF THE ROOTS OF A POLYNOMIAL EQUATION;129
12.1;Complex roots of a polynomial equation;130
12.2;The elementary symmetric functions pf the roots of a cubic equation;130
12.3;Sums of the powers of roots;134
13;CHAPTER 21. THE BINOMIAL THEOREM;138
13.1;The expansion of (a + x)n;138
13.2;The binomial theorem;142
14;CHAPTER 22. THE BINOMIAL SERIES;147
14.1;Limit sum;147
14.2;The binomial series;147
14.3;Relation with the binomial theorem;148
14.4;Expansion of (a + x)n;149
14.5;Partial fractions;152
14.6;Expansions;156
15;CHAPTER 23. NUMERICAL SOLUTION OF EQUATIONS;159
15.1;Outline of procedure;159
15.2;The first step;159
15.3;Linear interpolation;160
15.4;Newton's method;162
15.5;Trigonometric equations;164
15.6;Approximations;166
15.7;Recurrence relations;168
15.8;Expansions;170
16;CHAPTER 24. ANALYTICAL GEOMETRY;172
16.1;Vectors—a recapitulation;172
16.2;Vector spaces;174
16.3;Coordinates as vectors;174
16.4;Transfonnations;175
16.5;Euclidean space;176
16.6;Change of axes;178
16.7;Polar coordinates;179
16.8;Coordinate geometry;181
16.9;The parabola;182
16.10;The rectangular hyperbola;183
16.11;The circle;185
16.12;Orthogonal projection;186
16.13;The ellipse;186
16.14;Another form for the hyperbola;187
16.15;Polar coordinates;190
17;CHAPTER 25. THE ARGAND DIAGRAM;194
17.1;Lengths in the Argand diagram;196
17.2;Multiplication in the Argand diagram;197
17.3;Geometrical illustrations;198
17.4;Inversion;199
17.5;Complex equation of a line;199
17.6;Complex equations of a circle;200
17.7;Similarity;201
17.8;Conformal transformations;202
17.9;Polynomial equations;206
17.10;The equation xn= 1;207
18;CHAPTER 26. MATRICES AND DETERMINANTS;213
18.1;Transformations in a vector space;213
18.2;Successive transformations;214
18.3;Matrix—a definition;215
18.4;Addition of matrices;216
18.5;Equality of matrices;216
18.6;Multiplication by a scalar;217
18.7;Subtraction;217
18.8;The zero matrix;217
18.9;Product of two matrices;218
18.10;Geometrical applications;219
18.11;Matrix algebra;221
18.12;Unit matrices;221
18.13;Solution of linear equations;222
18.14;Elementary row operations;223
18.15;Reduction to the unit matrix;224
18.16;The inverse matrix;225
18.17;Determinants;226
18.18;Cofactors;228
18.19;Alien cofactors;229
18.20;Properties of determinants;229
18.21;Reduction of a determinant;231
18.22;Fonnation of the inverse matrix;233
18.23;Singular matrices;234
18.24;Quadratic forms;239
19;CHAPTER 27. THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS;243
19.1;The limit of a fmiction;243
19.2;Theorems on limits;244
19.3;Continuous functions;245
19.4;Definite integrals;246
19.5;Two important inequalities;248
19.6;The mean value theorem for integrals;249
19.7;The fundamental theorem;249
19.8;The function;250
19.9;L(x) is an increasing function of .;251
19.10;Numerical values of L(x);252
19.11;The value of L(x) when x is large;254
19.12;The value of L(x) when 0